102 IV. PRINCIPLE OF LEAST A(MON. GENERALIZED EQUAT. OF MOTION. 



velocities to be expressed in terms of the coordinates alone. To 

 effect this we make use of the equation of energy, 



from which 

 10) 



11) A =Jy2(h-W)Z r m r ds* 



In order to fix the ideas we may explicitly introduce a new 

 independent variable in the integral, supposing the equations of 

 motion to have been integrated and all the coordinates to be expressed 

 as functions of a single parameter g, which for example may be one 

 of the coordinates. That is for each value that is assigned to the 

 parameter q we suppose the position of every point in the system 

 completely known. 



ds 

 Writing now ds r = -~dq the integral is 



12) 



The proper statement of the principle of least action then is that 

 the variation of this integral vanishes, given the initial and final 

 configurations and the total constant energy. We have now com- 

 pletely get rid of the variable t, and are not embarassed by the 

 question whether its variation is zero or not. 



As the simplest possible example consider the case of a single 

 free particle acted on by no forces, then W = and the action is 



and the action is proportional to the distance traversed. 



If this is a minimum the path will be a straight line, the 

 principle of least action accords with Newton's first law. 



Suppose that the particle instead of being free is constrained to 

 lie on a given surface. The path described must then be an arc of 

 a shortest or geodesic line of the surface. The calculus of variations 



